a promblem about view frustum culling

i read graphics book, book says view frustum culling occurs between projection matrix transform and project divide.
it means that the point go to view frustum culling is nonhomogeneous coordinate.

then , for example it does culling by using Sutherland-Hodgeman algorithm, but i see some graphics books which describe this algorithm, it processes by the 3d coordinate. is it means that we need divide w to get the 3d coordinate? this way repeatly uses the action of projection divide in next step. maybe my thinking is wrong. but i don't know where is wrong.